DeepAI AI Chat
Log In Sign Up

Convergence of adaptive discontinuous Galerkin and C^0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

by   Ellya L. Kawecki, et al.

We prove the convergence of adaptive discontinuous Galerkin and C^0-interior penalty methods for fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including Poincaré and trace inequalities, a proof of density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results.


page 1

page 2

page 3

page 4


Convergence and optimality of an adaptive modified weak Galerkin finite element method

An adaptive modified weak Galerkin method (AmWG) for an elliptic problem...

On stabilizer-free weak Galerkin finite element methods on polytopal meshes

A stabilizing/penalty term is often used in finite element methods with ...

Convergence of an adaptive C^0-interior penalty Galerkin method for the biharmonic problem

We develop a basic convergence analysis for an adaptive C^0IPG method fo...

Uniform subspace correction preconditioners for discontinuous Galerkin methods with hp-refinement

In this paper, we develop subspace correction preconditioners for discon...

Implementation of hp-adaptive discontinuous finite element methods in Dune-Fem

In this paper we describe generic algorithms and data structures for the...